Narrow Results

Due to the essential difficulty of establishing an appropriate variational framework on a suitable working space, how to apply the critical point theory for showing the existence and multiplicity of periodic solutions of continuous-time difference equations remains a completely open problem. New ideas including gluing arguments are introduced in this work to overcome such a difficulty. This enables us to employ the critical point theory to construct uncountably many periodic solutions for a class of superlinear continuous-time difference equations without assuming symmetry properties on the nonlinear terms. The obtained solutions are piecewise differentiable in some cases, distinguishing continuous-time difference equations from ordinary differential equations qualitatively. To the best of our knowledge, this is the first time in the literature that the critical point theory has been used for such types of problems. Our work may open an avenue for studying discrete nonlinear systems with continuous time via the critical point theory.

We present difference equations for the simple biological aging model, which is called Stauffer model, and obtain the corresponding expression of the exact stationary solution. Considering numerically iterated solutions to those difference equations, we find that the result of iterated simulations is similar to the simulations of Stauffer model.

Looking for a handy and exact calculation for the equivalent resistance of an M×N resistor network is important, but difficult. In this paper, we present a standard and convenient approach to calculate the equivalent resistance of an M×N cobweb resistor network by applying multiple external current sources based on the nodal analysis in circuit theory and the recursion-transform (R-T) method. The test current source acts on different nodes in radial direction to obtain an analytical expression for the equivalent resistance between nodes of an M×N cobweb resistor network in radial direction. In our scheme, recalculations are not required to obtain the equivalent resistance between different radial nodes. We also discuss the influence of polygon sides of cobweb network and the ratio between two unit resistances on the equivalent resistance. The results show that, when the number of similar polygons M is given, with the increasing of the polygon sides and the ratio between two unit resistance, the equivalent resistances between two arbitrary radial nodes tend to a constant.

The capacitance between the origin and any other lattice site in an infinite square lattice of identical capacitors each of capacitance C is calculated. The method is generalized to infinite Simple Cubic (SC) lattice of identical capacitors each of capacitance C. We make use of the superposition principle and the symmetry of the infinite grid.

In this paper, we propose new difference equations which can generate the evolution rules of cellular automata.

In this paper, we design a cellular automaton and a discrete-time cellular neural network (DTCNN) using nonlinear passive memristors. They can perform a number of applications, such as logical operations, image processing operations, complex behaviors, higher brain functions, RSA algorithm, etc. By modifying the characteristics of nonlinear memristors, the memristor DTCNN can perform almost all functions of memristor cellular automaton. Furthermore, it can perform more than one function at the same time, that is, it allows multitasking.

The close relationship between chaos and cryptography makes chaotic encryption a natural candidate for secure communication and cryptography. In this manuscript, we prove that a class of maps that have been proposed as suitable for scrambling signals possess the property of sensitive dependence on initial conditions (s.d.i.c.) necessary for chaos and cryptography. Our result can also be used for generating other maps with s.d.i.c., through a suitable semiconjugacy between their input and output parts. Using the condition of semiconjugacy we also establish for this class of maps rigorous criteria for the existence and stability of their fixed points and limit cycles.

In this paper, we consider a one-dimensional difference equation with three parameters, the derivative of which at a fixed point has an eigenvalue $-1$ as the parameters are all zero. In the case that both nondegeneracy conditions of the flip bifurcation and the generalized flip bifurcation are not satisfied, by computing normal form, we give the nondegeneracy condition and transversality condition of the codimension-3 flip bifurcation. Moreover, by discussing the number of positive zeros of a cubic function in a neighborhood of the origin, we show the bifurcation scenario and give the parameter conditions, respectively, that the normal form of the equation possesses three 2-cycles, two 2-cycles, only one 2-cycle or none.

Dynamic behavior of a discrete-time prey–predator system with Leslie type is analyzed. The discrete mathematical model was obtained by applying the forward Euler scheme to its continuous-time counterpart. First, the local stability conditions of equilibrium point of this system are determined. Then, the conditions of existence for flip bifurcation and Neimark–Sacker bifurcation arising from this positive equilibrium point are investigated. More specifically, by choosing integral step size as a bifurcation parameter, these bifurcations are driven via center manifold theorem and normal form theory. Finally, numerical simulations are performed to support and extend the theoretical results. Analytical results show that an integral step size has a significant role on the dynamics of a discrete system. Numerical simulations support that enlarging the integral step size causes chaotic behavior.

This paper is concerned with chaotification of first-order partial difference equations. Two criteria of chaos for the difference equations with general controllers are established, and all the controlled systems are proved to be chaotic in the sense of Li–Yorke or of both Li–Yorke and Devaney by applying the coupled-expanding theory of general discrete dynamical systems. The controllers used in this paper can be easily constructed, facilitating the chaotification of first-order partial difference equations. For illustration, two illustrative examples are provided.

In this paper, we study the stability of the equilibrium point and investigate the periodicity properties of a class of difference equations $x_{n+1}=\frac{A_n{x}_n^3+B}{C{x}_{n-1}}$, $n\in \mathbb{N},$ whose parameters and initial values are positive. In particular, $A_n$ is a two-periodic coefficient with respect to n. Moreover, we depict the phase transition of orbits and analyze the deviation of the quasi-periodic orbits from some periodic orbits related to the arithmetic property of $A_n$, including Diophantine and Liouvillean numbers. We associate the response solution problem of differential equations with the fixed point problem of the area-preserving mapping related to multiperiodic difference equations, and pose some future work. The techniques are based on Moser’s twist mapping theorem, reversibility and some invariant structures embedded in the system.

The techniques of optimal control are applied to a validated blood circulation model of cardiopulmonary resuscitation (CPR), consisting of a system of seven difference equations. In this system, the nonhomogeneous forcing term is the externally applied chest pressure acting as the "control". We seek to maximize the blood flow, as measured by the pressure differences between the thoracic aorta and the superior vena cava. The new aspect in this application is that the control values from the two previous time steps are used to calculate the pressures (state variables) at the current time step. We prove the existence and uniqueness of the optimal control and provide a new CPR strategy, with increased blood flow. The characterization of the optimal control is given in terms of the solutions of the circulation model and of the corresponding adjoint system. The numerical results show a significant increase in the blood flow as compared with standard CPR.

In this paper, a discrete-time Ricker population model with the strong Allee effect is proposed, and its complex dynamic behavior is analyzed. First, the existence and asymptotic stability of the unique positive equilibrium are studied. Second, Neimark–Sacker and period-doubling bifurcations of this discrete model were carried out, and corresponding bifurcation conditions were obtained. Third, the pole placement method and the hybrid control strategy have been used to control the chaos produced by these bifurcations. Finally, we use MATLAB software to carry out some numerical simulations to analyze the rich dynamics of the system as well as to verify our theoretical results.

In this paper, we construct and analyze a mathematically reasonable and simplest population dynamics model based on Mark Granovetter’s idea for the spread of a matter (rumor, innovation, psychological state, etc.) in a population. The model is described by a one-dimensional difference equation. Individual threshold values with respect to the decision-making on the acceptance of a spreading matter are distributed throughout the population ranging from low (easily accepts it) to high (hardly accepts). Mathematical analysis on our model with some general threshold distributions (uniform; monotonically decreasing/increasing; unimodal) shows that a critical value necessarily exists for the initial frequency of acceptors. Only when the initial frequency of acceptors is beyond the critical, the matter eventually spreads over the population. Further, we give the mathematical results on how the equilibrium acceptor frequency depends on the nature of threshold distribution.

This paper is devoted to the investigation of the oscillation of a class of second-order nonlinear impulsive delay difference equations. Some interesting results are obtained, and some examples which illustrate that impulsive perturbations play a very important role in giving rise to oscillations of equations are also included.

An infinite homogeneous d-dimensional medium initially is at zero temperature, u=0. A heat impulse is applied at the origin, raising the temperature there to a value greater than a constant value u₀>0. The temperature at the origin then decays, and when it reaches u₀, another equal-sized heat impulse is applied at a normalized time τ₁=1. Subsequent equal-sized heat impulses are applied at the origin at the normalized times τ_n, n=2,3,…, when the temperature there has decayed to u₀. This sequence of normalized waiting times τ_n can be defined recursively by

A pair of linearly independent asymptotic solutions are constructed for the second-order linear difference equation

We derive uniform and non-uniform asymptotics of the Charlier polynomials by using difference equation methods alone. The Charlier polynomials are special in that they do not fit into the framework of the turning point theory, despite the fact that they are crucial in the Askey scheme. In this paper, asymptotic approximations are obtained, respectively, in the outside region, an intermediate region, and near the turning points. In particular, we obtain uniform asymptotic approximation at a pair of coalescing turning points with the aid of a local transformation. We also give a uniform approximation at the origin by applying the method of dominant balance and several matching techniques.

Human behavioral responses fundamentally influence the spread of infectious disease. In this paper, we study a discrete-time SIS epidemic process in random networks. Three forms of individual awareness, namely, local awareness, global awareness and contact awareness, are considered. The effect of awareness is to reduce the risk of infection. Based on the stability theory of matrix difference equation, we derive analytically the epidemic threshold. It is found that both local and contact awareness can raise the epidemic threshold, while the global awareness only decreases the epidemic prevalence. Our results are in line with a recent result using differential equation-based methods.

We consider two DNA sequences and compare both sequences. One of the crucial issues in bioinformatics is to measure the similarity of two DNA sequences. To this purpose one has to consider different alignments between both sequences. The number of alignments grows very rapidly with the length of the sequences. In this paper we give exact, explicit and computable formulas for the number of different possible alignments and for some classes of reduced alignments. We provide a new insight into the theory of DNA sequence alignment.